Compactly supported formulation of the sharp autocorrelation constant

Let $C_4$ be the smallest constant such that

$$\min_{t\in[0,1]}\int_{\mathbb{R}}f(x)f(t+x)\,dx\leq C_4\|f\|_1^2$$

for the function class specified in Theorem 4 of the source. The compactly supported formulation concerns functions $f\in L^1([-1/2,1/2])$. Compact-support conjecture for $C_4$.

$$C_4=\sup_{f\in L^1([-1/2,1/2])}\inf_{t\in[0,1]}\frac{|f\star f(t)|}{\|f\|_1^2}.$$

This conjecture would identify the best constant from the almost-compactly-supported problem with the supremum over compactly supported functions. The paper notes that an explicit compactly supported example gives a lower bound, while Theorem 4 only establishes a strict improvement over a previous upper bound; equality with the compact-support supremum remains open.